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magnetic moment

Magnetic moment is one of the core concepts in statics, used to describe the behavior of current loops or magnetic objects in a magnetic field:

1. Definition of magnetic moment

Formula: For a planar current loop, the magnitude of the magnetic moment m is the product of the current I and the loop area A, and the direction is 

determined by the right-hand rule:

M=I · A, where the direction of A is perpendicular to the plane of the loop.

Unit: Ampere per square meter (A · m ²).

Source: In addition to the current loop, the orbital motion and spin of microscopic particles (such as electrons) also generate magnetic moments.

2. The behavior of magnetic moment in an external magnetic field

Torque: When the magnetic moment is placed in the external magnetic field B, the torque received is:

T=m×B

This torque causes the magnetic moment to turn in the same direction as the external magnetic field.

Potential energy: The potential energy of a magnetic moment is determined by the relative direction of the magnetic moment and the magnetic field

U=-m·B

When the magnetic moment is aligned with the direction of the magnetic field, the potential energy is the lowest; Conversely, the highest.

3. Forces in non-uniform magnetic fields

The expression of force: In a non-uniform magnetic field, the force exerted on the magnetic moment is:

F=∇(m·B)

If the magnetic field is irrotational (∇× B=0), it is simplified as:

F=(m·∇)B

The magnetic moment tends to move towards the direction of magnetic field enhancement 

(when m is in the same direction as B).

4. Magnetic field generated by magnetic moment

Dipole field approximation: At a distance, the magnetic field generated by the magnetic moment is equivalent to the magnetic 

field of a magnetic dipole

$$B(r)=\frac{{\mu }_0}{4\mathrm{\Pi }}\left(\frac{3(m\cdot \hat{r})\cdot \hat{r}-m}{r^3}\right)$$

Where r ^ is the position unit vector and r is the distance.

5. Magnetic moment and material magnetism

Magnetization intensity: The magnetism of a substance is described by the magnetic moment per unit volume (magnetization 

intensity M). The total magnetic moment is the volume integral:

$$M_{\text{总 }}=\int Mvd$$

Magnetic field relationship: The magnetic field in matter satisfies

$$B={\mu }_0(H+M)<br>$$

Where H is the magnetic field strength.

6. Application and Extension

Angular momentum correlation: In classical theory, the magnetic moment of a rotating charged body is correlated with 

angular momentum L

$$M=\frac{q}{2m}L$$

In quantum mechanics, the demagnetization ratio factor of the electron spin magnetic moment is 2.

Experimental measurement: The magnetic moment can be measured by the force or vibration frequency in a 

non-uniform magnetic field.

summarize

Magnetic moment is the key to understanding the phenomenon of static magnetism, and its torque, potential energy, and 

force behavior in an external magnetic field explain the macroscopic phenomena of magnetic materials, such as magnetization

Hysteresis). Microscopic magnetic moments (such as electron spin) lay the foundation of material magnetism, including 

paramagnetism, ferromagnetism, and so on. Master the magnetic dipole field model

And the interaction with magnetic fields is the starting point for analyzing complex magnetic problems.